# A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

## Abstract

Herein, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the present method is displayed by numerical experiments on benchmark problems.

- Authors:

- Univ. of Maryland, College Park, MD (United States)

- Publication Date:

- Research Org.:
- Univ. of Maryland, College Park, MD (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); National Science Foundation (NSF)

- OSTI Identifier:
- 1598333

- Grant/Contract Number:
- [SC0009301; DMS-1418754]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- SIAM Journal on Scientific Computing

- Additional Journal Information:
- [ Journal Volume: 39; Journal Issue: 5]; Journal ID: ISSN 1064-8275

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; low-rank approximation; tensor format; stochastic Galerkin method; finite elements; GMRES; preconditioning; algebraic multigrid

### Citation Formats

```
Lee, Kookjin, and Elman, Howard C. A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations. United States: N. p., 2017.
Web. doi:10.1137/16M1075582.
```

```
Lee, Kookjin, & Elman, Howard C. A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations. United States. doi:10.1137/16M1075582.
```

```
Lee, Kookjin, and Elman, Howard C. Thu .
"A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations". United States. doi:10.1137/16M1075582. https://www.osti.gov/servlets/purl/1598333.
```

```
@article{osti_1598333,
```

title = {A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations},

author = {Lee, Kookjin and Elman, Howard C.},

abstractNote = {Herein, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the present method is displayed by numerical experiments on benchmark problems.},

doi = {10.1137/16M1075582},

journal = {SIAM Journal on Scientific Computing},

number = [5],

volume = [39],

place = {United States},

year = {2017},

month = {10}

}

*Citation information provided by*

Web of Science

Web of Science